## Integral Hopf-Galois structures on degree $$p^2$$ extensions of $$p$$-adic fields.(English)Zbl 0992.11065

This paper gives the definitive classification of Hopf Galois structures on valuation rings of Galois extensions of $$p$$-adic fields of degree $$p^2$$, $$p$$ prime.
Let $$L/K$$ be a finite, totally ramified Galois extension of $$p$$-adic fields of degree $$p^2$$ with Galois group $$G$$, and assume $$K$$ contains a primitive $$p$$th root of unity. Let $$e$$ be the absolute ramification index of $$K$$. Then there are $$p$$, respectively $$p^2$$ Hopf Galois structures on $$L/K$$ if $$G$$ is cyclic, respectively elementary abelian. Their structure depends at most on a field extension $$M$$ of degree $$p$$ over $$K$$. The author explicitly describes the corresponding $$K$$-Hopf algebras and their Hopf orders over $$\mathcal O_K$$, the valuation ring of $$K$$, extending work of C. Greither [Math. Z. 210, 37-68 (1992; Zbl 0737.11038)] and the reviewer [New York J. Math. 2, 86-102 (1996; Zbl 0884.11046)]. Using this he determines which of the Hopf orders are realizable, that is, can be the associated order of the valuation ring of a Galois extension of $$K$$. The results depend on $$G$$, two “valuation parameters”, $$i$$ and $$j$$, and a “unit parameter” $$v$$ (in $$M$$), as well as whether or not $$p = 2$$, and break up into seven cases corresponding to various subregions of the square $\{ (i,j) \mid 0 \leq i, j, \leq e/p \} .$ This is applied to determine which of the Hopf Galois structures on $$L$$ give rise to Hopf Galois structures on $$\mathcal O_L$$. The number of such Hopf Galois structures is either 1 or $$p$$.
The paper concludes by fitting into this landscape two classes of examples: Galois field extensions $$L/K$$ of degree $$p^2$$ that are classical Kummer extensions, and Kummer extensions for Lubin-Tate formal groups. The latter examples extend work of M. J. Taylor [J. Reine Angew. Math. 358, 97-103 (1985; Zbl 0582.12008)] and the author [Galois module structure and Kummer theory for Lubin-Tate formal groups, in Algebraic Number Theory and Diophantine Analysis, ed. by F. Halter-Koch and R. F. Tichy, de Gruyter, Berlin 55-67 (2000; Zbl 0958.11076)].

### MSC:

 11S23 Integral representations 11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers 11S31 Class field theory; $$p$$-adic formal groups 16W30 Hopf algebras (associative rings and algebras) (MSC2000)

### Citations:

Zbl 0737.11038; Zbl 0884.11046; Zbl 0582.12008; Zbl 0958.11076
Full Text:

### References:

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